This is knownhas been referred to as the Humanist Latin OrthographyNeo-Latin Orthography. An example of a grammar written with this type of orthography is An Introduction to the Latin Tongue by G. N. Wright.
The archaic genitive singular -ās of the ā-stems (the first declension) is marked with a circumflex: aurâs, Majâs, paterfamiliâs,
materfamiliâs, in imitation of Greek gen. sg. -ᾶς.
The genitive singular of the u-stems (the fourth declension) is marked with a circumflex, such as domûs, as if in imitation of the
Greek εὐτυχοῦς, Σαπφοῦς, αἰδοῦς or similar.
The nominative singular -ūs of the Greek names ending in -οῦς in the original, is also written with a circumflex: Trapezûs, Hierichûs.
The genitive plural ending -um (= -ōrum) was written with a circumflex mark, such as deûm (versus unmarked acc. sg. deum), in
imitation of the Greek gen. pl. ending -ῶν.
The genitive partitive of the personal pronouns nos, vos was marked with circumflex: nostrûm, vestrûm (versus unmarked acc. sg. masc. and
nom./acc. sg. neuter of the possessive nostrum, vestrum), in imitation
of Greek ἡµῶν, ὑµῶν.
The ablative singular of the ā-stems (the first declension) was marked with a circumflex: linguâ, Româ, versus unmarked nom. sg.
lingua, Roma. This particular case is likely to have been indirectly
influenced (or at least a posteriori justified) by Quintilian’s
teaching (1.7.3) about the usage of the apex mark.
Sometimes the explicative diacritic of the humanist Latin orthography coincided with the actual word accent: nostrâs, Samnîs,
which, according to Roman grammarians, could have been accented on the
last syllable.
The grave mark on the last syllable was often used to distinguish adverbs, prepositions, and some other, usually indeclinable
(Steenbakkers 1994, Burkard 2003), words: maximè (adverb) versus
maxime (vocative), cùm (conjunction) versus cum (preposition), verò
(conjunction) versus vero (dat./abl. sg.), quòd (conjunction) versus
quod (pronoun) etc. For the logic of such orthography, compare also
Quintilian’s mention (1.5.26) of the oxytonesis of auxiliary words,
and the Greek oxytonic grave, especially frequent in prepositions
(ἀνὰ, κατὰ, µετὰ, ὑπὸ etc).
Various humanist grammarians proposed also other orthographic rules to distinguish homographs, such as to mark the accent on the
penultimate syllable according to the Greek third mora rule, such as
to write lăbor and lābor respectively lábor and lâbor, or to write
pône (imperative of a verb) but ponè (preposition).
Sometimes contracted or syncopated word forms were written with a circumflex (dî = dii, nîl = nihil, mî = mihi, nôrat = noverat, etc.)
in imitation of the Greek contracted forms with the circumflex from
the first mora accent: νόος > νοῦς, φάος > φῶς etc.
The system of explicative diacritics in Latin orthography was abandoned between 18th and early 19th centuries.
Edit note: When I first posted this, I referred to this as the "Humanistic Latin Orthography" as Strockis called it. However, I later came across the following remark by Piet Steenbakkers:
"With hindsight we can perceive that it was humanistic research into
classical Latin which brought about the downfall of this unhistorical
orthography. The movement ad fontes and the scrutiny of ancient
inscriptions made humanists keenly aware of the examples they had to
follow in reforming their spelling and pronunciation of Latin."
It would, of course, be inaccurate to attribute this orthography to the humanists when they were the ones that led to its demise, so I felt it was better to use Steenbakkers' designation, "Neo-Latin Orthography."
